PHYSICAL REVIEW B VOLUME 48, NUMBER 21 1 DECEMBER 1993-I

Collective and single-particle excitations in a Hubbard antiferromagnet

Prasenjit Sen and Avinash SinghDepartment of Physics, Indian Institute of Technology, Kanpur 208016, India

(Received 1 June 1993)

The spectrum of spin-wave and single-particle excitations is studied in a half-filled-band Hubbard anti-ferromagnet for interaction strength values in the intermediate and weak-coupling regimes. The spin-wave-energy scale, which goes essentially as t /U in the strong- and intermediate-coupling regimes, ex-hibits a crossover and goes down with decreasing interaction strength in the weak-coupling regime. Thiscrossover is found to occur at U/t = 3.26, and it demarcates the Heisenberg behavior in the strong- andintermediate-coupling regimes from the spin-density-wave behavior in the weak-coupling regime. Thespin-wave-energy scale gets compressed by the charge gap in the weak-coupling regime, and consequent-

ly the spin-wave branch, which approaches the single-particle excitation band quite closely, never actual-

ly merges into it.

I. INTRODUCTION

The discovery of high-temperature superconductivity'in doped cuprate-based systems and the manifestation ofantiferromagnetic (AF) order in many of their parentcompounds have generated interest in low-dimensionalmagnetism in understanding the magnetically orderedand disordered ground states of these compounds.High-energy inelastic neutron scattering experiments in-dicate that in the magnetically ordered states undampedspin waves and in the magnetically disordered statesdamped spin waves are the basic low-energy magnetic ex-citations.

While the strong-coupling-equivalent of the Mott-Hubbard AF—the S=—,', quantum Heisenberg AF—has recently been studied quite intensively, it appearsthat it is the intermediate-coupling regime, wherein thecorrelation term ( U) is comparable to the free-particleband width ( W), which is of relevance for cuprates. Re-cent studies on cuprates indicate the presence of strongcovalency or hybridization eff'ects, leading to substantial-ly reduced sublattice magnetization. There have beenvery few investigations of the Hubbard AF in the inter-mediate (U —W) and weak- (U « W) coupling regimes.The sublattice magnetization and spin-wave velocity havebeen discussed by Hirsch and Tang, but the whole spin-wave spectrum was not studied and the numerical quan-tum Monte Carlo procedure faces limitations in theweak-coupling regime. Spin-Auctuation correction to thesublattice magnetization from the relevant self-energycorrection has been studied, however, the spin-wavepropagator for a general interaction strength was not dis-cussed, and apparently in the calculation the strong-coupling form of the spin-wave propagator was used.The series expansion method, in which the spin-wave en-ergy at the random phase approximation (RPA) level isobtained in the form of a series in powers of t /U, is oflittle quantitative value in the intermediate- and weak-coupling regimes, as with decreasing U the order towhich expansion needs to be carried out to providereasonable accuracy escalates rapidly.

We have obtained the spin-wave spectral properties ofthe Hubbard AF from the transverse spin susceptibilityevaluated at the RPA level. This many-body approachhas been used previously to obtain, in the strong-couplinglimit, the spin-wave energy, and also spin-fluctuationcorrections to sublattice magnetization, spin-wave energyand ground-state energy. The above approach has beenextended recently to the intermediate-coupling regimewherein the spin-wave energy was obtained in the form ofa series expansion in powers of t /U . There are in-teresting deviations in the spin-wave spectral propertiesfrom the strong-coupling limit, arising from theextended-range spin couplings generated in theintermediate-coupling regime. However, it becomesdif5cult to proceed with analytical calculation of spin-wave energies in the intermediate- and weak-coupling re-gimes, and in this paper we present results obtained bynumerically evaluating the spin-wave energies from thetransverse spin susceptibility.

Apart from the intrinsic information provided by thespin-wave energy in the intermediate- and weak-couplingregimes, this is also required in order to find the quantumspin-fluctuation corrections to the ground state energyand sublattice magnetization. The spin-wave density ofstates in the intermediate-coupling regime is also of in-terest with reference to the two-magnon Raman scatter-ing in cuprate antiferromagnets. ' Furthermore, thisstudy of the magnetic properties in the weak- andintermediate-coupling regimes provides for a comparisonwith exact numerical results when they become available.

We have obtained the spin-wave energy, 0&, for Qpoints on a 100X100 grid in Q space. We have con-sidered two cases, b, lt =3.0( Ult =7.0) and hit=0.7( Ult =2.7), representative of the intermediate- andweak-coupling regimes, and the spectral information ispresented in the form of spin-wave density of states,shown in Figs. 1 and 2. The logarithmic divergence inthe spin-wave density of states at energy 2J, due to thedegeneracy at the zone edge, is seen to be replaced by apeak structure in the intermediate- and weak-coupling re-gimes. The peak is seen to shift to lower energies and

0163-1829/93/48(21)/15792(5)/$06. 00 48 15 792 1993 The American Physical Society

48 COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS IN A. . . 15 793

5.00 1 .50

4.00— 1 . 25

3.00—

2.00—

1 . 00

0 .75

0.50

1 . 00— 0.25

0.00 0.2 5 0.50 0.75 1.00 1.25 0.00 0.50 1.00 1.50 2.00a„

2.50 3.00

FIG. 1. 1 Spin-wave density of states in the intermediate-coupling regime ( Ult =7).

FIG. 3. Spin-wave branch along the (0, 1)-direction in theweak-coupling regime ( U/t =2.7), and the lower band-edge ofsingle-particle excitations (2h).

simultaneously broaden with decreasing U. The max-imum spin-wave energy is found to occur at Q=(+ir, 0)or (0, +ir) in both the intermediate- and weak-couplingregimes, and the spin-wave branch along the (0, Q )-

direction is shown in Fig. 3. The spin-wave branch isseen to lie below the lower edge of the single-particle ex-citation band, and this is found to hold for all nonzero U.We have also studied the maximum spin-wave energy,which sets the spin-wave energy (SWE) scale, as a func-tion of Ult, and find that the maximum SWE peaks at3.26 (Fig. 4). This marks the crossover from Heisenbergto spin-density-wave behavior. With decreasing U/t, wealso find that the spin-wave branch approaches frombelow the single-particle excitation band quite closely inthe weak-coupling regime, without ever actually merginginto it. This is because when the spin-wave-energy scaleapproaches the charge gap, it gets compressed, and istherefore limited from above, by the decreasing chargegap.

We start with a brief review of the AF state to fix ournotation. In the AF state the Hubbard Hamiltonian atthe Hartree-Fock (HF) level can be represented in thetwo-sublattice basis as"

HHFko

[ (E —g )2 + e.2 )

1 /2 (2)

aqua =[1—bqt j' (3)

The sublattice or the staggered magnetization is equalto the absolute value of the local magnetization on anysite, and is obtained from the difference in the electronicdensities of the two spins,

in terms of the free-particle band energies,ck= 2t(co—sk„+cosk ), and the gap parameter b, .The eigenvalues yield the quasiparticle energies,Ek =+(b, +ez)'~, the plus and minus signs are for kstates lying outside and inside the Magnetic BrillouinZone (MBZ), respectively. The gap parameter is relatedto the sublattice magnetization I by 26=m U. Thequasiparticle wave functions are linear superposition ofplane waves on two sublattices A and B, the superposi-tion amplitudes, ak~, bk~, on the two sublattices formingthe eigenvector of Hk". The normalized superpositionamplitudes of l spin, for example, are given by

3.00

2.50—

2.00I

1 .50

1 .00

0.50—

0.00 0.25 0.50 0.75 1.00 1.2 5 1.50

2. 00

1 . 50

o1 . 00

I

C

0. 50

E

0. 00 2.00

/

//

/

s I i I ~ I I I I I I I I I I ~ I I I \ 1 I I I I I I I I I

4.00 6.00 8.00

U/t

2. 00

—1. 50

1.00

—0. 50

FIG. 2. Spin-wave density of states in the weak-coupling re-gime ( U/t =2.7).

FIG. 4. Dependence of maximum spin-wave energy (solidline) and 2h (dashed line) with U/t.

15 794 PRASENJIT SEN AND AVINASH SINGH

2, P 1 2, b,kt k1 N X

k b, +ek

1 1, 1

U N(4)

where the prime indicates that the sum is over all occu-pied states of the lower Hubbard band, i.e., k states lyinginside the Magnetic Brillouin Zone (MBZ). Since26 =—m U, this leads to the self-consistency conditionwhich determines the gap parameter, 6, and hence thesublattice magnetization in terms of the Hubbard interac-tion strength U:"

II. SPIN-WAVE AND SINGLE-PARTICLEEXCITATION ENERGIES

The spin-wave energies are obtained from the trans-verse spin susceptibility, g + (Q, Q), which at the RPAlevel, is given by

Xo(Q»)1 —Uyo(Q, Q)

+o,Q=

where y is the antiparallel-spin, particle-hole propaga-tor, evaluation of which, in terms of eigenvalues andeigenvectors of the HF Hamiltonian has been discussedearlier, " and in the two-sublattice basis, go(Q, Q) may beexpressed as

2

2a a k —Qo-e

xo(Q») =—g'a ko~ bko-~ a k —Qo e bk

koB ko.6 k —Qo 6 k —Qo eb2 bko e k —Qo.

1

Ek Q—Ek +O.A

where the sum is over all k states lying inside the MBZ,and over the two o. states, T and J, . The subscripts 6 and8 represent states in the upper and the lower Hubbardbands, respectively. Now, the spin-wave mode, which isa low-lying collective excitation in AF representing trans-verse spin fluctuations, shows up in the form of a pole inthe dynamical transverse spin susceptibility y (Q, Q).Hence from the RPA form, and the 2 X 2 structure of ppin the two-sublattice basis, it follows that for a given Qthe spin-wave energy, QQ, is a solution ofdet [ 1 —Uyo(Q, Q& ) ]=0. Now, owing to the spin-sublattice symmetries, we have ak =bk, so that oft'-

diagonal matrix elements in Eq. (6) are identical, andtherefore it follows that yp must be of the form"

Aq( —Q) Bq(Q )—I

8 (Q ) A (Q)Q Q(7)

Here, 3 and 8 are, as explicitly shown, odd and evenfunctions of Q, respectively and have the property that,for Q =0, 2 &+8&=0 for Q =1r and 0, respectively.That the leading order term in gp is U '1 follows fromthe self-consistency condition; this is required from thecontinuous spin rotational symmetry of the system, andensures a gapless mode. Thus the spin-wave energies aredetermined from the zeros of the functionfg(Q ):—Ag( —Q)Ag(Q) —Bq(Q )=0. fq(Q ) is aconvex and even function of 0, and in the strong-coupling limit, has the form Q, Q

—0, whereQg=2J'l/ 1 l g, with 7'g= 2 (cosg~ +cosgy ) 111 two-dimensional (2D), is the spin-wave energy in the strong-coupling limit.

To find the root QQ we adopt the iterative secantmethod. This requires two initial guesses close to thezero, yielding the slope of the function f, and then oneiterates rapidly towards the zero. In comparing f&(Q '

with zero, for the instant g„, we take a tolerance equal to

10 times the value of the function f evaluated at thepenultimate zero, Q& . Since Q is incremented in small

n —1

steps, this ensures that the tolerance is —10 times thechange in function f over the energy domain in whichsearch for the zero is conducted. In evaluating gp, the ksum is performed by summing over k points on a grid be-tween —~ and m in both directions. A 40X40 grid isfound to be adequate generally. However, we have goneup to a 120X 120 grid for the smallest b, /t(0. 45) caseconsidered. U is calculated for a given b, from Eq. (4) bydoing the k sum as stated before.

Spin-wave energies are obtained for 10000 Q points,from which the spin-wave density of states (SWDOS) isevaluated for two cases —b, =3.0 ( U/t =7) and 6=0.7( U/t =2.7), representative of the intermediate- andweak-coupling regimes. The spin-wave density of states,N(Q)=+&5(Q —Q&) is obtained, as usual, by approxi-mating the delta functions with Lorentzians,

N(Q) = lim—1 1

n~o vr Ng q (Q —Qq)1+112

where N& is the total number of Q points. 11 is chosen tobe of the order of the average level spacing in the regionof interest.

In Figs. 1 and 2 we show the spin-wave density ofstates for the two cases U/t =7 and U/t =2.7, represen-tative of the intermediate- and weak-coupling regimes.The obvious characteristic in both cases is that the diver-gence seen in the strong-coupling limit (Heisenberg case)at the upper band edge is removed and is replaced by apeak structure. With decreasing interaction strength thepeak broadens, and simultaneously shifts to lower ener-gies. In the intermediate-coupling case the density ofstates agrees well with the result obtained using the ex-pansion method in powers of t /U . However, in theweak-coupling case, we find additional structure near theupper band edge. The two curves clearly show that with

COLLECTIVE AND SINGLE-PARTICLE EXCITATIONS IN A. . . 15 795

decreasing interaction strength, progressively more spec-tral weight is transferred to the intermediate energydomain. This is a consequence of the deformation of theconstant spin-wave-energy surface caused by the extend-ed spin couplings generated with decreasing interactionstrength. The spin coupling between two spins becomesdependent upon their physical separation and not merelyon the number of hopping steps separating the two. Thisis discussed in more detail in the following paragraph.

In the strong-coupling limit, the spin-wave energy,Q&=2J+1 —

y&, has strong degeneracy at the straightlines in Q space joining the points (+m, 0), (0, +m) andsatisfying the relation, +Q, +Q =+~. The perfect fiat-ness of the spin-wave energy along these lines leads to thedivergence in the density of states. In the intermediate-and weak-coupling regimes this degeneracy is removed,with the maximum spin-wave energy occuring at the end-points of these lines, and the minimum at (+sr/2, +m/2).The same result had been obtained from the series expan-sion method, valid in the intermediate-coupling regime.This is a consequence of the extended-range spin cou-plings generated with decreasing U, with the couplingstrength decreasing with increasing distance betweenspins. The couplings between spins separated by twohopping steps, and hence in the same sublattice, [i.e. , thenext-nearest-neighbor (NNN) and next-next-nearest-neighbor (NNNN) couplings] are ferromagnetic in na-ture. This ferromagnetic coupling reduces the spin-waveenergy, and since this coupling is greater for the NNNpair, i.e., along the diagonal direction, the spin-wave en-ergy is reduced more for spin waves propagating alongthe (vr/2, m/2) direction. It is the local fatness of thedispersion near (+7r/2, +m/2) which leads to the peak inthe density of states at an energy below the upper bandedge. With decreasing U, the relative energy separationbetween modes with Q near the (+vr 2/, +sr/2) region andthose with Q near the (+m, O), (0, +sr) region increases,leading to the shifting-down in energy of the peak in thedensity of states.

The Q& vs Q surface is saddle-shaped around the fourpoints (+~/2, +~/2). 0& is convex in the four directionsalong (0,0) to (+sr/2, +sr/2), while it is concave along the(m /2, vr/2) to (vr, 0) and symmetrical directions. Thespin-wave dispersion is thus locally Bat around the eightpoints (+m/2, +m/2) and (+sr, O), (0, +sr). The doublephase space available around the former set, as comparedwith the latter set, is essentially the reason for the ap-proximately double density of states at the peak over thatat the upper band edge. In the long wavelength limit,Q ((I, the spin-wave energy is approximately linear in

Q, and the 0& surface is conical.We now discuss brieAy the single-particle excitations,

which are excitations across the charge gap. The single-particle excitation energy is the energy required to excitean electron form the lower Hubbard band to the upperHubbard band. Single-particle excitation modes corre-spond to poles in y + arising from singularities in theantiparallel-spin, particle-hole propagator, g0 itself.becomes singular when one of the energy denominators inEq. (6) is zero, i.e., 0=+Ek & E„. For each Q, —de-pending on k, there will be a range of single-particle exci-

tation energies. That the minimum is 2h for any Q canbe seen easily. From the expressions for the quasiparticleenergies, it follows that the minimum excitation energyoccurs when the pair k and k —Q both lie on the edge ofthe Magnetic Brillouin Zone. It is straightforward to seethat for any Q there is always a k on the MBZ boundarysuch that k —Q also lies on the MBZ boundary. We ex-pect this to hold for any simple continuous Fermi surfacein k space.

The spin-wave branch in the weak-coupling limit, to-gether with the lower edge of the single-particle excita-tion band (2b, ) is shown in Fig. 3. The Q dependence of0&, with Q„set to 0 is shown, and the behavior is quali-tatively identical to that in the strong-coupling limit.The Q dependence along the diagonal direction (Q, =Q~)is also similar. The highest energy in the spin-wavebranch [at (O, vr)] is slightly below the lower band edge ofsingle-particle excitations. This separation between thetwo edges is present for all nonzero U as discussed below,so that the spin-wave branch never seems to merge intothe single-particle excitation band.

In Fig. 4 we show the U dependence of the maximumspin-wave energy which sets the spin-wave energy scale.We have done calculations down to b, =0.45 ( U=2. 17).With decreasing 6 the k grid needs to be increasingly finein order to maintain numerical accuracy in the k sum,and this limits the numerical procedure. In the strong-coupling regime the spin-wave-energy scale goes as2J—:St /U, where J is the exchange energy of theequivalent spin- —,', nearest-neighbor Heisenberg AF. Inthe intermediate-coupling regime as well, this scale goesup roughly as t /U with decreasing U, so that the AF isessentially in the Heisenberg regime, albeit withextended-range spin couplings. However, the spin-waveenergy scale shows a crossover behavior and this scalestarts going down below U/t =3.26. This crossovermarks the onset of a regime fundamentally di6'erent fromthe Heisenberg regime. Since the sublattice magnetiza-tion also falls drastically at around this value of U, weidentify this regime as the spin-density-wave regime.

In Fig. 4 we have also plotted 26, the minimumsingle-particle excitation energy. It is seen that with de-creasing U/t the separation between the maximum spin-wave energy and the minimum single-particle excitationenergy, 25 progressively decreases and, from the trend,seems to vanish only as U~O. Thus the spin-charge sep-aration, while becoming marginal, is still maintained, andthe highest spin excitation energy consistently lies belowthe lowest charge-excitation energy. The spin-wavebranch never actually merges into the single-particle exci-tation band. As the gap for single-particle excitationsgoes down with decreasing U, it seems to compress thespin-wave energy scale, so as to keep the spin-wavebranch out of the single-particle band. The Heisenbergand spin-density-wave regimes are therefore basicallycharacterized, respectively, by the conditions Q&&&2b,when spin-charge separation is complete, and Q& ~ 2b„when this separation is marginal.

We note that the maximum spin-wave energy is limitedfrom above by the single-particle excitation gap of 2h.This may be understood in the following way. The spin-

15 796 PRASENJIT SEN AND AVINASH SINGH

wave-energy scale is essentially set by the denominator ofthe linear in 0 term in go(Q, A). In the strong-couplinglimit, this denominator is nothing but the exchange ener-

gy, J. In the intermediate-coupling regime as well, thisdenominator is essentially independent of frequency.However, with decreasing gap parameter, as the frequen-cy 0 approaches 2A from below, this denominator be-comes strongly frequency dependent, and is stronglysuppressed with increasing frequency. Thus whenQ, ~26, this spin-wave energy scale becomes dynamicallysuppressed, and gets limited by the charge gap 2A.Hence, with decreasing U, both 2A and the maximumspin-wave energy go to zero as U~O.

III. CONCLUSIONS

In conclusion, we have studied the spin-wave spectralproperties of the Hubbard antiferromagnet in theintermediate- and weak-coupling regimes, by exactlyevaluating the spin-wave energy at the RPA level. Wefind that the Hubbard antiferromagnet exhibits a richvariety of behavior, going through a crossover from aHeisenberg AF in the strong- and intermediate-couplingregimes, to a spin-density-wave system in the weak-coupling limit. With decreasing U, the spin-wave energyscale increases approximately at t /U in the strong- andintermediate-coupling regimes, however, in the weak-coupling regime, it gets compressed by the decreasingcharge gap, and is forced to go down in parallel with 2A.That the spin-wave-energy scale is limited from above bythe charge gap is due to a dynamical suppression opera-tive when the frequency approaches the charge gap.

Therefore, in the Hubbard antiferromagnet, the spin-wave branch never actually merges into the single-particle excitation band, but approaches it, more andmore closely, with decreasing U. We believe this featureto survive even in absence of nesting and to hold for ageneric antiferromagnet, and also when quantum spinfIuctuation effects are included. Now, when Auctuationsare included, the charge gap gets narrowed, whereas thespin-wave energies are enhanced. The exact maximumspin-wave energy must lie above the RPA value andbelow the HF 26 value, and therefore, in view of thecloseness of the two limits, we can say that the Auctua-tion effects are insignificantly small in the weak-couplingregime. We have also investigated the spin-wave densityof states in the intermediate- and weak-coupling regimes,and find significant differences from the strong-couplingcase. The presence of a peak structure which broadens,and moves down in frequency with decreasing U, isrelevant for a complete discussion of two-magnon Ramanscattering in the Hubbard antiferromagnet in theintermediate- and weak-coupling regimes. The peakstructure resembles the broadening effect of spin-wave in-teraction, and implies that the intrinsic broadening in thenoninteracting spin-wave density of states be incorporat-ed as well.

ACKNOWLEDGMENTS

Computations were performed on the Convex 220computer system in the IIT-Kanpur Computer Centre.P.S. wishes to thank P. Shandilya for his help in the com-putations.

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